Liouville-type theorems for subharmonic functions on complete manifolds

Applying the generalized maximum principle, a Liouvilletype theorem of subharmonic functions on complete Riemannian manifolds is shown and a Liouville-type differential inequality on properly immersed complete submanifolds is given. Project supported by the National Natural Science Foundation of China and K.C. Wong Educational Fund     

Tautness for riemannian foliations on non-compact manifolds

For a riemannian foliation on a closed manifold M, it is known that is taut (i.e. the leaves are minimal submanifolds) if and only if the (tautness) class defined by the mean curvature form (relatively to a suitable riemannian metric μ) is zero (cf. álvarez in Ann Global Anal Geom 10:179–194, 1992). In the transversally orientable case, tautness is equivalent to the non-vanishing of the top basic cohomology group , where (cf. Masa in Comment Math Helv 67:17–27, 1992). By the Poincaré Duality (cf. Kamber et and Tondeur in Astérisque 18:458–471, 1984) this last condition is equivalent to the non-vanishing of the basic twisted cohomology group , when M is oriented. When M is not compact, the tautness class is not even defined in general. In this work, we recover the previous study and results for a particular case of riemannian foliations on non compact manifolds: the regular part of a singular riemannian foliation on a compact manifold (CERF). J. I. Royo Prieto was partially supported by EHU06/05, by a PostGrant from the Basque Government and by the MCyT of the Spanish Government. R. Wolak was partially supported by the KBN grant 2PO3A 021 25.     

Diffusion processes on complete riemannian manifolds

ＤＩＦＦＵＳＩＯＮＰＲＯＣＥＳＳＥＳＯＮＣＯＭＰＬＥＴＥＲＩＥＭＡＮＮＩＡＮＭＡＮＩＦＯＬＤＳＱＩＡＮＺＨＯＮＧＭＩＮ（钱忠民）（ＤｅｐａｒｔｍｅｎｔｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ,ＳｈａｎｇｈａｉＩｎｓｔｉｔｕｔｅｏｆＲａｉｌｗａｙＴｅｃ...     

A x -operator on complete riemannian manifolds

In this paper we give a generalisation of Kostant’s Theorem about theA _x -operator associated to a Killing vector fieldX on a compact Riemannian manifold. Kostant proved (see [6], [5] or [7]) that in a compact Riemannian manifold, the (1, 1) skew-symmetric operatorA _x =L _x −≡ _x associated to a Killing vector fieldX lies in the holonomy algebra at each point. We prove that in a complete non-compact Riemannian manifold (M, g) theA _x -operator associated to a Killing vector field, with finite global norm, lies in the holonomy algebra at each point. Finally we give examples of Killing vector fields with infinite global norms on non-flat manifolds such thatA _x does not lie in the holonomy algebra at any point.     

A symplectic integrator for riemannian manifolds

Summary The configuration spaces of mechanical systems usually support Riemannian metrics which have explicitly solvable geodesic flows and parallel transport operators. While not of primary interest, such metrics can be used to generate integration algorithms by using the known parallel transport to evolve points in velocity phase space. This paper is dedicated to the memory of Juan C. Simo This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan Simo.     

Curvature-orbits and locally homogeneous riemannian manifolds

We prove that it is possible to associate to each infinitesimal model on a Euclidean vector space V a locally homogeneous Riemannian manifold. As an application, we characterize, in the space of the algebraic curvature tensors on V, the orbits which can occur for locally homogeneous Riemannian manifolds.     

Hessian equations on compact non-negatively curved riemannian manifolds

On a compact connected riemannian manifold, we partly extend to hessian equations an existence result proved for Monge-Ampère equations. Non-negative curvature is required for a priori estimates. Received: 13 July 2001 / Accepted: 25 October 2001 / Published online: 29 April 2002 The author is supported by the CNRS